Consider a sequence $x_1,$ $x_2,$ $x_3,$ $\dots$ defined by
\begin{align*}
x_1 &= \sqrt[3]{3}, \\
x_2 &= (\sqrt[3]{3})^{\sqrt[3]{3}},
\end{align*}and in general,
\[x_n = (x_{n - 1})^{\sqrt[3]{3}}\]for $n > 1.$   What is the smallest value of $n$ for which $x_n$ is an integer?
Answer: We have that
\[x_3 = (\sqrt[3]{3})^{\sqrt[3]{3}})^{\sqrt[3]{3}} = (\sqrt[3]{3})^{\sqrt[3]{9}},\]and
\[x_4 = (\sqrt[3]{3})^{\sqrt[9]{3}})^{\sqrt[3]{3}} = (\sqrt[3]{3})^{\sqrt[3]{27}} = (\sqrt[3]{3})^3 = 3,\]so the smallest such $n$ is $\boxed{4}.$